Then it is easy to see that \(\lbrace U_n\rbrace _{n=0}^\infty \)  is a
complete orthonormal system for the weighted \(L^2\)  space \(L^2(I,w)\) ,
where \(w(t):=(1-t^2)^\frac{d-1}{2}\) . Let \(\mathbf {B}^d\)  be the unit
ball in \(\mathbf {R}^d\) , \(\mathbf {S}^{d-1}\)  be the unit sphere in
\(\mathbf {R}^d\)  and \(\mathcal {P}_n\)  be the set of algebraic polynomials
of degree not larger than \(n\)  defined on \(\mathbf {B}^d\) . Denote by
\(d\omega _{d-1}\)  the aero element of \(\mathbf {S}^{d-1}\) . Then \(\Omega _{d-1}:=\int _{\mathbf {S}^{d-1}}d\omega _{d-1}=\frac{2\pi ^{\frac{d}{2}}}{\Gamma (\frac{d}{2})}.\) 
The
following important properties of \(U_n\)  are established in
[1]}.
